Author Topic: Goursat Problem  (Read 930 times)

Jingxuan Zhang

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Goursat Problem
« on: January 23, 2018, 10:05:09 PM »
I am approaching homework 3 question 2. It is very similar to the question worked out in lecture and the one in text but I must admit I am still not very content with my understanding. In particular,

1. should we not expect the solution is only determined up to an unknown function in two region;
2. How exactly do we translate the region $x>c|t|$ to the characteristic? My current idea is it will be come the first quadrant ($\eta~\xi$), but this is merely empirical.

Victor Ivrii

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Re: Goursat Problem
« Reply #1 on: January 24, 2018, 01:31:53 PM »
You can characteristic coordinates $(\xi,\eta)$; alternatively you can use the general solution of the wave equation, Anyway, $u$ is sought in the quadrant $\{(x,t)\colon x>c|t|\}$. Draw it!

Jingxuan Zhang

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Re: Goursat Problem
« Reply #2 on: January 27, 2018, 10:53:46 AM »
Indeed I drew that right-ward triangle but what should the domain of independence be? should that be a triangle with its tip point to the right (so that the two sides intersect the two information)?
« Last Edit: January 27, 2018, 10:56:34 AM by Jingxuan Zhang »

Victor Ivrii

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Re: Goursat Problem
« Reply #3 on: January 28, 2018, 04:15:03 AM »
Domain of dependence. You can figure it out, solving the problem

Jingxuan Zhang

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Re: Goursat Problem
« Reply #4 on: January 30, 2018, 01:13:04 PM »
Solved. Only till yesterday did I find out that domain of dependence really depends on the particular problem.

Ioana Nedelcu

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Re: Goursat Problem
« Reply #5 on: January 31, 2018, 12:08:14 AM »
My take on it: it is a linear combination of the solution to the homogeneous equations given the boundary conditions (1) and the inhomogeneous solution (2)

(2) is simple, using the Duhamel integral or change of variables to the characteristic curves: $$ \frac{-1}{4c^2} \int _{0}^{t} \int_{x-c(t-t')}^{x+c(t-t')} f(x',t')\,dx'dt' $$

(1) is a bit confusing because of the intervals: $$ g(t) = \phi(2ct) + \psi(0),  t>0 $$ $$ h(t) = \phi(0) + \psi(2ct),  t<0 $$ Do we use an odd extension/ method of reflection for the functions h and g so we can solve them?

Victor Ivrii

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Re: Goursat Problem
« Reply #6 on: January 31, 2018, 11:45:19 AM »
Just do it in characteristic coordinates. If $f=0$ it is easy. Assume that $f\ne 0$ but boundary values are $0$ (Linearity). What would be formula for $u(\xi,\eta)$? What domain of integration? (Rectangle). And in $(x,t)$? What will be Jacobian?